Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. Passes through \(\left(\frac{1}{4},-\frac{2}{3}\right)\) and \(\left(-\frac{1}{2}, \frac{1}{3}\right)\)

Step-by-Step Solution

Answer

The standard equation of the line that passes through the points \(\left(\frac{1}{4},-\frac{2}{3}\right)\) and \(\left(-\frac{1}{2}, \frac{1}{3}\right)\) is \(2(\frac{1}{3} - y) = -2m\left(-\frac{1}{2} - x\right)\). Note that \(m\) is the slope calculated in the first step.

Step 1: Calculate the Slope

The slope of a line passing through the points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\) can be found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's substitute given points into the formula: \(m = \frac{\frac{1}{3}-\left(-\frac{2}{3}\right)}{-\frac{1}{2}-\frac{1}{4}}\)

Step 2: Calculate the y-intercept

After calculating the slope, use the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\). Substituting one of the given points (let's choose \(\left(-\frac{1}{2}, \frac{1}{3}\right)\)) and the calculated slope \(m\) into this formula, we can find the y-intercept: \(\frac{1}{3} - y = m\left(-\frac{1}{2} - x\right)\)

Step 3: Transform the equation into standard form

Next, rewrite the equation from the slope-intercept form to standard form. The standard form of a line is \(Ax + By = C\), where A, B, and C are integers, and A and B are not both zero. Multiply both sides by -2 to clear the fractions: \(2(\frac{1}{3} - y) = -2m\left(-\frac{1}{2} - x\right)\)

Key Concepts

Standard Form of a Line

Understanding the standard form of a line is fundamental in algebra. It is usually expressed as
\[ Ax + By = C \]
where A, B, and C are integers, and A should be non-negative. If A is zero, the line is horizontal, and if B is zero, the line is vertical. The standard form is useful because it can easily be rearranged to yield the slope-intercept form \( y=mx+b \)
or used to find the x and y-intercepts analytically. For example, setting y to zero yields x-intercept and setting x to zero yields y-intercept. In the above exercise, converting the point-slope form to standard form involves algebraic manipulation to clear fractions and arrange the terms properly, which results in integers that correspond to A, B, and C in the standard form equation.

Calculating Slope

Slope is a measure of the rate at which a line inclines, or declines, which is crucial when graphing linear equations. It is calculated by finding the ratio of the vertical change to the horizontal change between two distinct points on the line. The slope is usually denoted as
\( m \)
and the formula to calculate it is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In the context of the above exercise, the slope was found using the coordinates of the two given points. Ensuring to use the precise values in the correct order when subtracting ensures an accurate calculation of the line's slope. Once the slope is known, it becomes the 'm' in the slope-intercept form of a line, which is instrumental for describing the line's direction.

Point-Slope Form

The point-slope form is an intuitive way to write the equation of a line when you know the slope and one point on the line. It's written as:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) represents the coordinates of the known point, and \(m\) is the slope of the line. This form is extremely useful because it clearly shows the line's slope and one specific point through which it passes. From the point-slope form, you can derive both the slope-intercept form and the standard form by algebraic manipulation, which was exemplified in the exercise when substituting the calculated slope and the chosen point to express the line's equation.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's significant because it helps to visually place the line on a coordinate grid. In algebraic terms, it's represented as the 'b' in the slope-intercept form \( y=mx+b \). To find the y-intercept from an equation, set \(x=0\) and solve for \(y\). In the step-by-step solution provided, the y-intercept was being calculated as part of the process to derive the standard form equation of the line. Identifying the y-intercept helps to understand how the line will behave in graph form and anchors the line on the graph at a specific point along the y-axis.